Might help to think about two circles of equal radius at first, or even try it physically with two coins of the same denomination.
Suppose the fixed one is on the left and the moving one starts off exactly to the right of it and rolls around over the top of the fixed one. When it has laid off half its circumference (against half the circumference of the fixed one), it is now exactly to the left of the fixed one, but it is the same way up as it originally was and has made one revolution about its centre. Then when it continues around the bottom of the fixed one, it makes another revolution, so when it gets back to where it was, it has made two revolutions, despite the fact that the circumferences are the same in this case. So here, though the "obvious" answer is 1, the correct answer is 2.
Kevin
-----Original Message-----
From: Ted Harding [mailto:[log in to unmask]]
Sent: 16 June 2011 14:31
To: [log in to unmask]
Subject: Re: Delighted to be corrected
Restating the question as originally posted:
> The question was as follows:
> "A circle of radius 1cm rolls round the outside a circle of 3cm.
> The point A is the initial point of contact of the two circles.
> How many revolutions does the smaller circle make before it
> returns to point A?
> 2? 3? 4.5? 6?"
I take it as implied here that the larger circle remains fixed,
and the point A (initial point of contact) likewise remains
fixed on the larger circle.
Just to be clear, let the smaller circle carry a label B at the
initial point of cintact, and that B remains fixed with respect
to the smaller circle as it rolls.
Then (as I stated before) the number of revolutions that the
smaller circle makes before it returns to point A (i.e. until
B is again coincident with A) is 4. Not what one immediately
thinks!
There are two ways of seeing this.
[A] Imagine that instead of rolling, the smaller circle is
slid around the larger with the point B (fixed w.r.to the
smaller circle) always in contact with the larger, so no
rolling. Then, in performing this motion, the smaller circle
carries out 1 revolution. Now go back to the start, and
this time roll it. Then in addition to the 1 revolution just
described, the smaller circle performs 3 additional revolutions
due to the rolling. Total: 4.
[B1] In the initial state (say B in contact with A at top of
larger circle), cut the larger circle at A and unwind the
part that goes off to the left until it is a horizontal
straight line, left-hand end A in contact with the point B
on the smaller circle. Now roll the smaller circle to the
other end (i.e. rightwards). Then it weill have performed
3 clockwise revolutions.
[B2] Now restore the straight line to the circle by reversing
the procedure of cutting and unwinding in [B1], keeping the
smaller circle in rigid contact with the endpoint. Doing so
rotates the smaller circle by an additional clockwise rotation.
Total: 4.
Ted.
On 16-Jun-11 13:08:13, Jane Galbraith wrote:
> Clearly I need to brush up my circular geometry. Along a straight line
> the
> answer would have been 1 (or 3) but when A on the little circle first
> touches the bigger circle the little one will have moved 1/3 way round
> the
> big circle and when it gets back to the starting point it will have
> gone
> all the way round the big circle. Could I argue that relative to the
> larger circle...?
> Thank you to Maya,
> Jane Galbraith
>
> ---------------------------- Original Message
> ----------------------------
> Subject: Re: Oh those exam questions!
> From: "maja zaloznik" <[log in to unmask]>
> Date: Thu, June 16, 2011 12:52 pm
> To: [log in to unmask]
> ------------------------------------------------------------------------
> --
>
> Well actually, if A were only marked on the smaller circle the answer
> would
> have been 1 1/3.
> Does that help?
> maja.
>
> On 16 June 2011 12:17, Jane Galbraith <[log in to unmask]> wrote:
>
>> Sorry, if the RADII are 1 and 3cm and the point of contact, A, is
>> marked
>> on the larger circle then why isn't the answer 3? Of course, if A were
>> only marked on the smaller circle the answer would have been 1.
>>
>> Jane Galbraith
>> PS But I'm glad some people know a Poisson can't be negative.
>>
>> > It was said that in the USA a multiple choice maths question was
>> > answered
>> > by
>> > over 100,000 examinees of whom only one had the temerity to point
>> > out
>> that
>> > all the answers were wrong. The question was as follows:
>> > "A circle of radius 1cm rolls round the outside a circle of 3cm.
>> > The
>>
>
>
>> > point A is the initial point of contact of the two circles. How
>> > many
>> > revolutions does the smaller circle make before it returns to point
>> > A?
>> 2?
>> > 3? 4.5? 6?"
>> >
>> > A Cambridge statistical examiner was criticised for asking
>> > candidates to
>> > prove that both the sum and the difference of two Poisson variables
>> > has
>> > the
>> > Poisson distribution. This time the bolder entrants did comment
>> > that
>> > Poisson variables cannot be negative.
>> >
>> > Gavin Ross
>> >
>> > ----- Original Message -----
>> > From: "Ted Harding" <[log in to unmask]>
>> > To: <[log in to unmask]>
>> > Sent: Thursday, June 09, 2011 8:22 AM
>> > Subject: Oh those exam questions!
>> >
>> >
>> >> Greetings All!
>> >> Perhaps a bit off-topic for us, but I reckon it's a
>> >> good place to ask the question.
>> >>
>> >> I've been repeatedly hearing/seeing teasing news items
>> >> about the "unanswerable" AS-level exam questions set by
>> >> different exam boards:
>> >>
>> >> A maths question which was impossible to answer
>> >> because not enough information was given (OCR)
>> >>
>> >> A business studies question which did not give enough
>> >> information (AQA)
>> >>
>> >> A multiple-choice biology question which gave several
>> >> possible answers but not the correct one (Edexcel)
>> >>
>> >> (descriptions quoted from the BBC News website).
>> >>
>> >> I say "teasing" because I'd love to know the details of
>> >> these questions, just to see how "goofy" they really are.
>> >> But nowhere have I come across citations of the actual
>> >> questions themselves, nor any detail much more specific
>> >> than the above.
>> >>
>> >> So can anyone provide full citations, or pointers to where
>> >> they can be found?
>> >>
>> >> My interest is aroused because I've set (and seen) a good
>> >> few exam questions in my time, and therefore have an eye
>> >> for the "warning signs" of a dodgy question. I'd like to
>> >> be able to see how blatant these were in the above cases!
>> >>
>> >> With thanks, and best wishes to all.
>> >> And apologies if I have set an unanswerable question.
>> >>
>> >> Ted.
>> >>
>> >> --------------------------------------------------------------------
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Fax-to-email: +44 (0)870 094 0861
Date: 16-Jun-11 Time: 14:30:42
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